/** This library calculates the correlation between items in lists. */



/** This calculates the autocorrelation of a single numeric list, that is, it
    identifies the periods at which sequences appear to repeat.  The result
    is a sorted list containing pairs of [offset, correlation] with the terms
    of strongest correlation being first in the list.

    If you just want a single most likely autcorrelation period, it is found at
       result@0@0

    see correlationTest.frink for an example of using it.
*/
autocorrelation[list] :=
{
   results = new array
   size = length[list]
   average = sum[list] / size

   // Scale the list around its average
   normalizedList = new array[size+1]
   denom = 0
   for i = 1 to size
   {
      term = list@(i-1) - average
      normalizedList@i = term
      denom = denom + term^2
   }

   for offset=1 to size
   {
      sum = 0
      for i = 1 to size-offset
         sum = sum + normalizedList@i * normalizedList@(i+offset)

      results.push[[offset, sum/denom]]
   }

   return sort[results, byColumn[1]]
}


/** This is the partial autocorrelation of a series.  The partial
    autocorrelation is useful because it eliminates the effects of smaller
    offsets.  For example, the partial autocorrelation at offset k is the
    autocorrelation not accounted for by offsets 1 through k-1.  That means
    that, for example, if a sequence has a perfect autocorrelation with a
    offset of 5, this will *not* display any partial autocorrelation at
    10, 15, etc.

    For equation:
    https://support.minitab.com/en-us/minitab/20/help-and-how-to/statistical-modeling/time-series/how-to/autocorrelation/methods-and-formulas/methods-and-formulas/
*/
partialAutocorrelation[list] :=
{
   results = new array
   size = length[list]
   average = sum[list] / size

   // Scale the list around its average
   normalizedList = new array[size+1]
   denom = 0
   for i = 1 to size
   {
      term = list@(i-1) - average
      normalizedList@i = term
      denom = denom + term^2
   }

   for offset=1 to size
   {
      sum = 0
      for t = offset+1 to size-offset
         sum = sum + normalizedList@(t-offset) * normalizedList@(t)

      results.push[[offset, sum/denom]]
   }

   return sort[results, byColumn[1]]
}